Thermoacoustic streaming on a sphere

Abstract

The fundamental problem of thermoacoustic streaming on a rigid sphere in a strong standing acoustic field has been treated analytically. The sphere radius (a) is taken to be large compared with the displacement amplitude (A) of fluid oscillations, and yet small on the scale of the radian wavelength (λ/2π) of the sound field. Only the high frequency limit is considered here, for which the Stokes oscillatory boundary layer thickness (δv) is much smaller than the sphere radius (a), and the streaming effects in the boundary layer are most pronounced. It is found that the phased interaction of the first–order harmonic quantities in the boundary layer is capable of introducing a second–order time–averaged temperature distribution, in addition to the well–known second–order time–averaged fluid motion. The associated steady temperature gradients cause localized heating and cooling variations over the surface of the sphere, whose net result is always a mean heating of the sphere. The role of little–known second–order thermodynamic moduli is pointed out, which, however, do not contribute to this phenomenon for the case of an ideal gas host fluid. Results for this time–averaged thermal effect are presented and discussed with reference to a possible novel application to the acoustic heating of small particles using ultrasonic frequencies for which heat fluxes of O (1) kW m–2 can be achieved.